Inner amenability of locally compact quantum groups

Abstract

We initiate a study of inner amenability for a locally compact quantum group G in the sense of Kustermans-Vaes. We show that all amenable and co-amenable locally compact quantum groups are inner amenable. We then show that inner amenability of G is equivalent to the existence of certain functionals associated to characters on L1(G). For co-amenable locally compact quantum groups, we introduce and study strict inner amenability and its relation to the extension of the co-unit ε from C0(G) to L ∞(G). We then obtain a number of equivalent statements describing strict inner amenability of G and the existence of certain means on subspaces of L∞(G) such as LUC(G), RUC(G) and UC(G). Finally, we offer a characterization of strict inner amenability in terms of a fixed point property for L1(G)-modules. © 2013 World Scientific Publishing Company.